USING TRIANGULAR FUZZY NUMBERS FOR MEASURING QUALITY OF SERVICE FROM THE CLIENT S PERSPECTIVE IN THE HOTEL INDUSTRY

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1 USING TRIANGULAR FUZZY NUMBERS FOR MEASURING QUALITY OF SERVICE FROM THE CLIENT S PERSPECTIVE IN THE HOTEL INDUSTRY Olimpia Iuliana, Ban; Nicoleta, Bugnar University of Oradea, Str. UniversităŃii, 0087 Oradea,, tel oban@uoradea.ro, nbugnar@uoradea.ro hotel, service quality, aggregation, triangular fuzzy number The answers to the questionnaires applied in the tourist industry contain a certain level of un-determination. In the paper we are trying to overcome this impediment by using the triangular fuzzy number. We transform into triangular fuzzy numbers the tourists answers, we operate with them, we obtain as results triangular fuzzy numbers also, and when needed we defuzzify the result. The results obtained can make the managers aware of the position of the unit they manage and can signal to them the activities that they should improve in order to gain a better position in the clients opinion. INTRODUCTION We are processing the data obtained as a result of the questionnaire implementation to the tourists accommodated at three of the hotels in the Neptun resort. We are using a linguistic scale with five stages for each of the important criteria taken into consideration. For the answers obtained contain a certain level of un-determination, the configuration with the help of fuzzy numbers is proved suitable. We are using the triangular fuzzy numbers (as the simplest fuzzy numbers to be able to operate with the tourists answers. In order to make a synthesis of the results for each criterion and each hotel, we aggregate the fuzzy numbers using the arithmetic average operator in its fuzzified form. We obtain a synthesis indicator for each hotel aggregating the results already obtained for the criteria considered separately. This aggregation is weighted by the tourists answers to another question of the questionnaire regarding the importance of the criteria considered, independently of the unit where they are accommodated now. The result obtained is a triangular fuzzy number. We obtain a hierarchy of the hotels observed, defuzzifying the results and comparing the real numbers found. In the second section there are presented in a simple form the notions necessary for the paper regarding the triangular fuzzy numbers, as well as some recent works where the fuzzy mathematics has been used to configure in the filed of service quality. The third section contains the work hypotheses and the primary processing of the results obtained. The fourth section is the main section of the paper. Here are obtained the results regarding the quality of services provided in these three hotels, either in the form of triangular fuzzy numbers or in the form of classical numbers, using simple aggregation operators. These allow us to order the hotels, both for each criterion separately, as well as synthetically, taking into account the totality of criteria considered. TRIANGULAR FUZZY NUMBERS Fuzzy numbers allow us to model the linguistic expressions appeared in different scientific areas mainly because of dependence on human judgement, finite resolution of measuring instruments or finite representation of numbers in computers. The service quality is a composite of various criteria, among them many are difficult to measure exactly such that the linguistic terms like poor, fair, good, very good or somewhat important, important, very important, extremely important are used. Under these 68

2 circumstances, fuzzy set theory (fuzzy numbers, particularly was already proved to be an adequate frame to overcome some linguistic problems ;[see J. M. Benitez, J. C. Martin, C. Roman, 007; S.-H. Tsaur, T.-Y. Chang, C.-H. Yen, 00 and C. H. Yeh, Y.-L. Kuo, 003]. Let X be a nonempty set. A fuzzy set A in X, is a set A {( x, µ A ( x : x X } in X is a set, where µ A : X [ 0,] is called the membership function of A and µ A ( x indicates the value of truth of the statement that the element x belongs to set A [see L. Zadeh, 965]. Triangular fuzzy numbers are particular fuzzy sets in R [see D. Dubois, H. Prade, 978] completely characterized by three real numbers t, t, t3 such that t < t < t3 with the membership function x t, if t x t t t x t3 µ T ( x, if t x t3 t t3 0, otherwise. T t, t t a triangular fuzzy number as above. We denote by (, 3 The addition of two triangular fuzzy numbers T ( t, t, t3 and ' ( t', t ', t3' T T ' ( t + t', t + t', t + t 3 ' 3 T is defined b and the scalar multiplication of T by real number λ > 0 is defined by λ T ( λt, λt, λt3. A well-known aggregation operator (in fact the fuzzified form of the weighted arithmetic mean of the triangular fuzzy numbers T,...,Tn is defined by f MAPω ( T,..., Tn ω T... ωn Tn, ( ω with ω, the addition and scalar multiplication defined where,..., [ 0,] ω n n i i above. Often it is need to convert a triangular fuzzy number into a crisp real number, that is to locate the best non-fuzzy value to the initial fuzzy value. There exist several available methods, in this paper we use the expected value [D. Dubois, H. Prade, 987; S. Heilpern, T t, t t then its expected value is introduced by 99]. If (, 3 t + t + t3 EV ( T. ( Many authors have proposed different methods for ranking fuzzy numbers, particularly triangular fuzzy numbers [see G. Bortolan, R. Degani, 985] for the review of methods. In the present paper we use, for its simplicity especially, the order denoted by p and defined by [R. R. Yager, 98] T ( T EV (. p T ' EV T ' 69

3 WORK HYPOTHESES AND PRIMARY RESULTS - We have used a questionnaire applied during 8 July August 005 to a number of 56 tourists accommodated [see O. Ban, A. Tomescu, 005] at three hotels in the Neptun resort [see O. Ban, A. Tomescu, 005]; - We have taken into consideration the answers to the question:. Which criteria regarding the quality of services (from the list of those in Table are important for you? Please mark them from to 0, 0 points for the most important. The scores can be repeated to take into account in this way the tourists opinions regarding the quality of services. We obtained the weights ω i in Table, which were calculated in the article [O. Ban, 007]. - To question 5. How do you appreciate these aspects at this hotel? (from the list of those in Table Give points from to 0, point for the worst complying with the criterion. We have replaced the scores obtained with linguistic scales to be able to illustrate the use of triangular fuzzy numbers in processing the answers, these containing a certain level of un-determination. The answers to question 5 are synthesised in Table. The numbers written in the table represent the number of tourists accommodated in one of the three hotels (R RomanŃa, T Terra, C Clăbucet who gave the answers in the upper part of the table. We add that the level written in the table are extremely important, very important, important, to a certain extent, to a little extent are just generic, these could be modified varying with the criterion considered, keeping the number of stages (5 and the descending order (for example, from extremely important to to a certain extent. 70

4 TableTourists answers to questionnaire H.RomanŃa (R- tourists; H.Terra (T-6 tourists; H.Clăbucet (C-9 tourists. Quality criteria Extremely Very Important To a To a important important certain extent little extent R T C R T C R T C R T C R T C. Interior and exterior aspect Modernity of the building 3. Modern equipment facilities. Ambient Cleaning state Using computerbased systems for registering tourists and taking 7. Well trained personnel 8. Personnel s availability to serve Promptness in serving clients 0. Communication skills. Kindness Enthusiasm and good-humored 3. Capacity of dealing and solving problems. Capacity of task fulfilling SYNTHESIS RESULTS To operate the answers provided by the tourists and to achieve the objective had in view, that to order the hotels varying with each criterion and then reported to the totality of criteria, we are assigning for each possible stage a triangular fuzzy number. We are making a standard choice which is represented in Figure. This means that to extremely important answer corresponds the triangular fuzzy number u 5(7.5, 0,0, to the very important answer corresponds the triangular fuzzy number u (5.0, 7.5, 0, to the important answer the triangular fuzzy number u 3(.5,5.0,7.5, to the a certain extent answer the triangular fuzzy number u (0,.5,5.0,and to the little extent answer the triangular fuzzy number u (0,0,.5. 7

5 Figure The triangular fuzzy number for the characteristics: to a little extent ((a, thin line, to a certain extent ((b, callous line, important ((c, thick line, very important ((d, double line and extremely de important ((e, triple line For instance, (d in Figure should be interpreted as follows: a tourist who answers very important to one of the questions gives 7,5 with degree, gives 6,5 with degree 0,5 and gives 5 with degree 0. Or we can say that the degree of membership of the mark 7,5 to the very important answer is, that of the mark 6,5 is 0,5, and that of the mark 5 is 0., from the list of those considered, calculating the arithmetic average of the fuzzy numbers obtained as answers provided by the tourists, that is using for criterion k the formula (formula ( with the weights considered equal with /n We evaluate each hotel from the point of view of criterion k, k {,..., } ( n u n u n3 u3 n u n5 u5, n n i, i,,3,,5 notes the number of tourists providing an answer corresponding the where { } triangular fuzzy number n n n n3 n n5 u i to the question regarding the k quality criterion, and For example, the evaluation of the Romanta hotel from the point of view of Interior and exterior aspect criterion is calculated as follows: ( 6 u5 6 u 3 u3 5 u u (( 5,60,60 ( 30,5,60 ( 7.5,5,.5 ( 0,.5,5 ( 0,0,.5 ( 8.5,3.5,70 ( 3.93,6.3,8.0. Similarly the evaluation of the Terra hotel by the tourist is calculated from the point of view of Enthusiasm and good-humored criterion: 7

6 6 6 6 ( 5 u u u 3 u u 5 (( 37.5,50,50 ( 0,30,0 ( 5,0,5 ( 0,7.5,5 ( 0,0,5 ( 6.5,97.5,5 ( 3.9,6.09, Similarly, the other data obtained from the tourists for each criterion considered and for each hotel are synthesised. The results are presented in Table. Table Synthesis of answers under the form of triangular fuzzy numbers and classical form Criteria and their importa nce RomanŃa H. Terra H. Clăbucet H. fuzzy real fuzzy real fuzzy real (3.93,6.3, (.06,6.5, (.08,6.5, (3.95,5., (3.9,6.5,8. 6. (3.9,5.6, (.05,6.07, (.,6.7, (3.8,5.79, (6.3,8.8, (5.7,7.97, (5.79,8.6, (7.50,0, (7.03,9.37, (7.50,0, (3.69,6.07, (3.9,6.09, (.08,6.5, (5.,7.7, (.06,6.09, (5.6,7.76, (6.90,9.0, (6.87,9.38, (7.,9.7, (7.6,9.76,0 9.9 (6.7,9., (7.,9.7, (6.3,8.8, (5.9,8., (6.5,8.95, (6.9,8.69, (7.9,9.69,0 9. (6.5,8.95, (.6,6.90, (3.9,6.09, (.7,6.7, (6.3,8.93,9. 8. (.06,6., (6.3,8.8, (5.7,8., (.38,6.7, (5.9,8., To be able to compare the situation of the hotels relatively to a certain criterion, we defuzzified the fuzzy numbers obtained using the expectance value formula (. In the case of criterion.interior and exterior aspect for the Romanta hotel we obtain: 73

7 , and in the case of criterion.enthusiasm and good-humored for the Terra hotel: Similarly we obtain the other numbers contained in Table. We notice that we cannot classify the three hotels globally because for different criteria we obtain different hierarchies. For example, for criterion.modernity of the building the order (ascending is Terra, Romanta, Clabucet, for criterion.ambient the order (ascending is Romanta, Terra, Clabucet, and for criterion 0.Communication skills the order (ascending is Clabucet, Romanta, Terra. Moreover, for a correct hierarchy we should take into account the importance given by the tourists to different criteria, that is the weights of the criteria presented in the second column in Table. To measure synthetically the quality of services in each of the three hotels, from the tourists perspective, when the questionnaire was applied and based on the quality criteria suggested, indicator noted S Romanta, STerra, SClabucet, we will calculate the fuzzy weighted arithmetic average of the results obtained for each hotel. For Romanta hotel we obtain: S Romanta ( 3.93,6.3, ( 3.95,5., (.05,6.07,7.6 ( 6.3,8.8, ( 7.50,0, ( 3.69,6.07,7.7 ( 5.,7.7, ( 6.90,9.0, ( 7.6,9.76,0 ( 6.3,8.8, ( 6.9,8.69, (.6,6.90,8.5 ( 6.3,8.93, ( 5.7,8., ( 5.677,8.038, , and similarly for Terra and Clabucet hotels: S S Terra Clabucet ( 5.568,7.608, ( 5.688,8.0767, If we want to compare the results according to the order introduced in the section Fuzzy triangular number, we must calculate the expectancy values using the formula (. We obtain: EV EV ( S Romanta ( STerra ( S EV Clabucet and also the order (descending of complying with the quality criteria Clabucet, Romanta, Terra, With the mention that the difference between Clabucet and Romanta hotels is extremely small (below 0.0 on a scale with 0 units or, in percentage terms, below.%. 7

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